In recent years, there have been proposed and developed various engine cylinder cutoff control technologies. One such engine cylinder cutoff control device has been disclosed in Japanese Patent Provisional Publication No. 10-82334 (hereinafter is referred to as “JP10-82334”). In the engine cylinder cutoff control device or the engine cylinder resting control device as disclosed in JP10-82334, one of a pair of cylinder cutoff cams (or a pair of cylinder resting cams) is provided at one side of a pair of intake and exhaust valve operating cams, while the other of the cylinder cutoff cam pair is provided at the other side of the intake and exhaust valve operating cams. Each of the cylinder cutoff cams is designed to have only a base circle. An intake-valve rocker arm and a pair of cylinder cutoff rocker arms located on both sides of the intake-valve rocker arm are rockably supported on an intake-valve rocker shaft. Simultaneously, an exhaust-valve rocker arm and a pair of cylinder cutoff rocker arms located on both sides of the exhaust-valve rocker arm are rockably supported on an exhaust-valve rocker shaft. The tips of the two intake-valve cylinder cutoff rocker arms are abutted-engagement with respective intake-valve-stem ends, while the tips of the two exhaust-valve cylinder cutoff rocker arms are abutted-engagement with respective exhaust-valve-stem ends. Also provided is a coupling/uncoupling switching mechanism, comprised of a plurality of plunger pistons, disposed in an interior space of a common cylindrical bore coaxially penetrating the rockably fitted base portion of the intake-valve rocker arm and the rockably fitted base portions of the cylinder cutoff rocker arms. Depending on engine/vehicle operating conditions, the intake-valve rocker arm and the intake-valve cylinder cutoff rocker arms are selectively coupled with each other or uncoupled from each other and simultaneously the exhaust-valve rocker arm and the exhaust-valve cylinder cutoff rocker arms are selectively coupled with each other or uncoupled from each other, by means of the coupling/uncoupling switching mechanism comprised of the plunger pistons, such that the engine operates at either one of the full cylinder operating mode and the partly-inactive cylinder operating mode.